On the quadrature exactness in hyperinterpolation
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This paper investigates the role of quadrature exactness in the approximation scheme of hyperinterpolation. Constructing a hyperinterpolant of degree n requires an m-point positive-weight quadrature rule with exactness degree 2n. Aided by the Marcinkiewicz–Zygmund inequality, we affirm that when the required exactness degree 2n is relaxed to n+k with 0<k≤ n, the L^2 norm of the hyperinterpolation operator is bounded by a constant independent of n. The resulting scheme is convergent as n→∞ if k is positively correlated to n. Thus, the family of candidate quadrature rules for constructing hyperinterpolants can be significantly enriched, and the number of quadrature points can be considerably reduced. As a potential cost, this relaxation may slow the convergence rate of hyperinterpolation in terms of the reduced degrees of quadrature exactness.
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